STAT10x9.8.05.DataRelationships

STAT10x9.8.05.DataRelationships - Data Relationships Today...

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10 15 20 10 20 30 40 50 60 70 80 Diameter Volume Data Relationships Today: describing relationship of two quantitative variables : 0. Scatterplots 1. Correlation 2. Regression Visualizing Relationships : Scatterplots o Plot two variables simultaneously o Put one variable on horizontal axis, other variable on vertical axis o Plot data pairs – for each observation, plot the value of one variable vs. the value of the other variable Example : Height and Diameter of 31 Black Cherry Trees in Alleghany Forest. Here is a sample of the data and a Scatterplot : Diameter Height Volume 8.3 70 10.3 8.6 65 10.3 8.8 63 10.2 10.5 72 16.4 10.7 81 18.8 10.8 83 19.7 STAT 101-106 Introduction to Statistics 49

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Scatterplot Notation : The Horizontal axis is ALWAYS called the X axis . The Vertical axis is ALWAYS called the Y axis. Association between Variables Some values of the first variable seem associated with particular values of the second variable. Example : Factors associated with Fertility Rate (based on 2000 World Poverty Survey) Positive Association STAT 101-106 Introduction to Statistics 50 0 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 7 Female Illiteracy Fertility Rate 40 50 60 70 80 1 2 3 4 5 6 7 Life Expectancy 40000 30000 20000 10000 0 7 6 5 4 3 2 1 GNI per capita Fertility Rates X Y Note : Association talks about direction of relationship, not the Negative Association
Sample Correlation 3. Measures the strength of the linear relationship between two variables. 4. Denoted by r 5. Value is between –1 and +1 Weak positive correlation Strong positive correlation (near zero) (near one) r = 0.06 r = 0.99 Moderate negative correlation Strong negative correlation r = - 0.52 r = - 0.96 STAT 101-106 Introduction to Statistics 51 -2 -1 0 1 2 3 -2 -1 0 1 2 -2 -1 0 1 2 3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 -2 -1 0 1 2 3

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Definition of correlation o First standardize variables: use z -scores ! y i y x i x s y y z s x x z i i - = - = and i.e. How many SD’s is each observation above or below the mean for each variable? This is just a change of units – same picture! Original and z-scores for the cherry tree data : o Second, multiply and average : o Algebra shows this is the same thing : STAT 101-106 Introduction to Statistics 52 ( 29 = - = n i y x i i z z n r 1 1 1 - - - - = 2 2 ) ( ) ( ) )( ( y y x x y y x x r i i i i
o Algebra also shows this is a dimensionless number between –1 and +1 (try this if you like!) STAT 101-106 Introduction to Statistics 53

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Idea of Correlation Formula What is the value of - - y i x i s y y s x x ? Sample Correlation in MINITAB : Stat Basic Statistics Correlation. Enter the two variables of interest. STAT 101-106
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STAT10x9.8.05.DataRelationships - Data Relationships Today...

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